Increasing the Output Length of Zero-Error Dispersers

• Authors: Ariel Gabizon;Ronen Shaltiel
• Venue: APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
• Year: 2008

Let $\cal C$ be a class of probability distributions over a finite set 茂戮驴. A function is a disperserfor $\cal C$ with entropy thresholdkand errorif for any distribution Xin $\cal C$ such that Xgives positive probability to at least 2kelements we have that the distribution D(X) gives positive probability to at least elements. A long line of research is devoted to giving explicit (that is polynomial time computable) dispersers (and related objects called "extractors") for various classes of distributions while trying to maximize mas a function of k.In this paper we are interested in explicitly constructing zero-error dispersers(that is dispersers with error ). For several interesting classes of distributions there are explicit constructions in the literature of zero-error dispersers with "small" output length mand we give improved constructions that achieve "large" output length, namely m= 茂戮驴(k).We achieve this by developing a general technique to improve the output length of zero-error dispersers (namely, to transform a disperser with short output length into one with large output length). This strategy works for several classes of sources and is inspired by a transformation that improves the output length of extractors (which was given in [29] building on earlier work by [15]). Nevertheless, we stress that our techniques are different than those of [29] and in particular give non-trivial results in the errorless case.Using our approach we construct improved zero-error dispersers for the class of 2-sources. More precisely, we show that for any constant 茂戮驴 0 there is a constant 茂戮驴 0 such that for sufficiently large nthere is a poly-time computable function such that for any two independent distributions X1,X2over such that both of them support at least 2茂戮驴nelements we get that the output distribution D(X1,X2) has full support. This improves the output length of previous constructions by [2] and has applications in Ramsey Theory and in constructing certain data structures [13].We also use our techniques to give explicit constructions of zero-error dispersers for bit-fixing sources and affine sources over polynomially large fields. These constructions improve the best known explicit constructions due to [26,14] and achieve m= 茂戮驴(k) for bit-fixing sources and m= k茂戮驴 o(k) for affine sources.